By V. S. Varadarajan
ISBN-10: 082180989X
ISBN-13: 9780821809891
This article bargains a unique account of Indian paintings in diophantine equations through the sixth via twelfth centuries and Italian paintings on recommendations of cubic and biquadratic equations from the eleventh via sixteenth centuries. the amount strains the ancient improvement of algebra and the speculation of equations from precedent days to the start of recent algebra, outlining a few smooth subject matters akin to the elemental theorem of algebra, Clifford algebras, and quarternions. it truly is aimed at undergraduates who've no historical past in calculus.
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Example text
Henc e deduc e tha t (f r > 0 and j9 i > po > 0 . (Hin t : Th e formul a fo r p i i s a straightforwar d calculation . Nex t not e that ( v A r + p o ) / | ^ o | — l / ( v ^ ~ P o ) > 1 , provin g th e secon d assertion . 7lo| < X\ < v-/V , and the n C = q\. Sho w fro m th e abov e inequalit y tha t C > 0. ) 6. Sho w fro m th e formula e ((i -+ - 1)) tha t fo r al l Z > 0 , Pi+1 = a i+lPi + P i -1 > tft+ 1 = a i + i f t + (fc_ i where Xi+i + re t ^i+i — —j i — with th e convention s q$ = l , p _ i = 1 , #—i = 0 , CLQ Pi+i Rul e II I (whe n p i s odd ) o f Brahmagupt a no w gives (P , Q; 1) where P = ^1 1 x 1 1 8 = 649 , Q = ^3 x 120 = 1 8 0 so tha t X = 649 , Y = 180 is a positiv e integra l solutio n o f X2 - 1 3Y 2 = 1 2. A 7" = 61 . A calculation show s tha t (39)2 - 6 1 x 5 2 = - 4 so tha t (39,5;-4) Rule I V (wit h p odd ) no w gives , afte r a quic k calculatio n usin g a han d calculator , (P,Q;1), wher e 1522 x 1 52 4 - 2 P = 1 52 3 x = 2 1 76631 904 9 Q = ^3 9 x 5 x 1 52 2 x 1 52 4 = 2261 5398 0 Thus X = 1 76631 9049 , Y = 2261 5398 0 is a positiv e integra l solutio n o f X2 - 61 Y 2 = 1 D E S C R I P T I O N O F TH E CAKRAVAL A W e shal l no w describ e th e algorithm o f cakravala, i n a for m tha t i s a varian t o f th e origina l on e a s describe d in [W ] (se e p . Gel'fond , an d b y Th . Schneider , i n 1 934 . Such question s severel y ta x th e resource s o f moder n mathematics . Fo r difference s between algebrai c an d transcendenta l number s i n th e closenes s t o whic h the y ca n be approximate d b y rationa l numbers , se e [HW ] [S]. NOTES A N D EXERCISE S 1. Verif y th e inequalitie s use d i n th e Archimedes ' approximation . 2. Prov e th e trigonometri c identitie s use d i n th e Archimedes ' approximation . To obtai n a n insigh t int o th e approximation s fo r th e squar e root s i n Archimedes' s work , se e [H-A ] (Introduction, pp . Algebra in Ancient and Modern Times by V. S. Varadarajan
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